- Email: [email protected]
Synchronizing chaotic systems up to an arbitrary scaling matrix via a single signal
Applied Mathematics and Computation 218 (2012) 6118–6124
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Synchronizing chaotic systems up to an arbitrary scaling matrix via a single signal Giuseppe Grassi a,⇑, Damon A. Miller b a b
Dipartimento Ingegneria Innovazione, Università del Salento, 73100 Lecce, Italy Department of Electrical and Computer Engineering, Western Michigan University, Kalamazoo, MI 49008, USA
a r t i c l e
i n f o
Keywords: Chaos synchronization Observer-based synchronization Chaotic systems with attractor scaling Projective synchronization Full state hybrid projective synchronization
a b s t r a c t This paper introduces a novel type of synchronization, where two chaotic systems synchronize up to an arbitrary scaling matrix. In particular, each drive system state synchronizes with a linear combination of response system states by using a single synchronizing signal. The proposed observer-based method exploits a theorem that assures asymptotic synchronization for a wide class of continuous-time chaotic (hyperchaotic) systems. Two examples, involving Rössler’s system and a hyperchaotic oscillator, show that the proposed technique is a general framework to achieve any type of synchronization defined to date. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The concept of chaos synchronization has attracted considerable attention in recent years. In 1991 Carroll and Pecora [1] showed that the dynamics of a drive system and of a driven subsystem (response system) become synchronized if the Lyapunov exponents of the response system are less then zero. Since their work, several methods for obtaining synchronized systems (including neural networks and complex systems) have been developed [2–14]. Among these, some researchers have focused their attention on the synchronization of hyperchaotic systems using a scalar synchronizing signal [4–10]. A particular type of chaos synchronization was introduced in 1999 [15]. In this type of synchronization (called projective synchronization) the response system states are scaled replicas of the drive system states [16]. Since the scaling factor was not predictable in [15], some control methods have been successively proposed, with the aim to choose any desired scaling factor [16–20]. A variation of project synchronization is the idea of full state hybrid projective synchronization, which has been recently developed in [21–24]. In this type of synchronization the scaling factor can be different for each state variable. This enables different kinds of synchronization to be achieved. For example, in Ref. [21] anti-synchronization and complete synchronization are simultaneously achieved in a symmetrical coordinate subspace and a normal coordinate subspace, respectively. In [22] adaptive control is exploited to achieve the full state hybrid projective synchronization of a new 4D hyperchaotic system with unknown parameters. Additionally, in [23,24] an active control scheme is utilized to achieve full state hybrid projective synchronization in some hyperchaotic continuous-time systems. The approaches in [21–24] share the drawback that synchronization is achieved using several control signals, while communications based on projective synchronization require a scalar synchronizing signal [25]. This paper presents a further contribution to the topic of synchronization by introducing a novel scheme, where two chaotic systems synchronize up to an arbitrary scaling matrix. Specifically, each drive system state synchronizes with a linear
⇑ Corresponding author. E-mail address: [email protected] (G. Grassi). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.097
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combination of response system states by using a single synchronizing signal. The proposed approach proves to be a general framework, which includes (as a particular case) any type of synchronization defined to date. The paper is organized as follows. Section 2 introduces the proposed observer-based approach, along with a theorem that guarantees synchronization can be achieved for any arbitrary scaling matrix under certain broad conditions. A remarkable feature is that the proposed type of synchronization is achievable via a single signal for a wide class of continuous-time chaotic systems. Finally, Section 3 illustrates two examples, involving Rössler’s system [26] and the hyperchaotic oscillator in [11], showing that identical, projective, anti-phase and full state hybrid projective synchronization are all achievable using the conceived synchronization scheme. 2. Synchronizing chaotic systems up to an arbitrary scaling matrix Definition 1. Given the continuous-time drive and response systems described by the n-dimensional state vectors x(t) and ^ðtÞ, respectively, these two systems are synchronized up to an arbitrary scaling matrix when, for an initial condition x ^ð0Þ, there x is a matrix
2
3
a11 a12 a1n a22 a2n 7 7
6a 21 a¼6 6 4
72R 5
nn
ð1Þ
an1 an2 ann such that:
! n X kei ðtÞk ¼ xi ðtÞ aij ^xj ðtÞ ! 0 as t ! 1; j¼1
i ¼ 1; 2; . . . n:
ð2Þ
The condition (2) means that each drive system variable xi(t) synchronizes with a linear combination of response system variables ðai1 ^ x1 ðtÞ þ ai2 ^ x2 ðtÞ þ ain ^ xn ðtÞÞ; i ¼ 1; 2; . . . n. Referring to the systems to be synchronized, this paper focuses on the class of dynamical systems defined by the following state and output equations, respectively:
_ xðtÞ ¼ AxðtÞ þ bf ðxðtÞÞ þ c;
ð3Þ
yðtÞ ¼ f ðxðtÞÞ þ kxðtÞ;
ð4Þ
n1
nn
n1
n1
n
where xðtÞ 2 R is the state vector, A 2 R ; b 2 R and c 2 R are constant matrices, f : R ! R is a scalar nonlinear function, y(t) is a scalar synchronizing output, and k 2 R1n is a gain vector to be determined. As pointed out in [7], the class described by (3) includes several well-known chaotic and hyperchaotic systems, such as Chua’s circuit [2], Rössler’s system [26], the Matsumoto–Chua–Kobayashi circuit [27], the hyperchaotic oscillator in [11], the higher dimensional Chua’s circuit in [28] and the circuit for generating multi-scroll attractors in [29]. Given the drive system (3), the observed-based approach enables the state x(t) to be reconstructed by a response system using a scalar function of the states (4). By utilizing the fact that the response system need not be a copy of the drive system in order to obtain synchronized dynamics (see [8,11]), a theorem is now proposed, where synchronization up to an arbitrary scaling matrix is achievable via a single signal. Theorem 1. If the matrix
b Ab A2 b An1 b
ð5Þ
is full rank, then the nonlinear dynamical system
^ðtÞÞ þ a1 c þ a1 bðyðtÞ y ^_ ðtÞ ¼ a1 Aax ^ðtÞ þ a1 bf ðx ^ðtÞÞ; x ^ ^ ^ yðtÞ ¼ f ðxðtÞÞ þ kaxðtÞ
ð6Þ ð7Þ
and the drive system (3) synchronize for any arbitrary scaling matrix a, provided that the eigenvalues of the matrix
½A bk
ð8Þ
are placed in the left-half plane by suitable k. Proof. First of all, it is worth noting that the error (2) can be written in matrix form as
^ðtÞÞk ! 0 keðtÞk ¼ kðxðtÞ ax By taking into account the error defined in (2), it follows that:
ð9Þ
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_ _ ax ^ðtÞ bf ðx ^ðtÞÞ c bðyðtÞ y ^_ ðtÞÞ ¼ AxðtÞ þ bf ðxðtÞÞ þ c Aax ^ðtÞÞ eðtÞ ¼ ðxðtÞ ^ðtÞÞ þ bf ðxðtÞÞ bf ðx ^ðtÞÞ bðf ðxðtÞÞ þ kxðtÞ f ðx ^ðtÞÞ kax ^ðtÞÞ ¼ AðxðtÞ ax ^ðtÞÞ bkðxðtÞ ax ^ðtÞÞ ¼ Ae bke ¼ ðA bkÞeðtÞ ¼ AðxðtÞ ax
ð10Þ
_ If matrix (5) is full rank, the error system eðtÞ ¼ ðA bkÞeðtÞ can be globally asymptotically stabilized at the origin by suit^ð0Þ and for any arbitrary scaling matrix a. Thus, according to (2) able k [7,8]. This means that ke(t)k ? 0 for t ? 1, for any x each drive system state synchronizes with a linear combination of response system states via the scalar synchronizing signal (4). Remark 1. The proposed scheme represents a general framework to achieve any type of synchronization defined to date. Namely, when a11 = a22 = = ann = 1 and aij = 0 i – j, identical (complete) synchronization is achieved, whereas for a11 = a22 = = ann = 1 and aij = 0 i – j, anti-phase synchronization is obtained. Moreover, when a11 = a22 = = ann = a and aij = 0 i – j, the projective synchronization as originally proposed in [15] is obtained. On the other hand, when a11 – a22 – – ann and aij = 0 i – j, the scaling matrix a is diagonal with different entries, and therefore full state hybrid projective synchronization as defined in [24] is achieved. Remark 2. The proposed observer-based method shares the same advantages illustrated in [8], i.e., it is rigorous and systematic, it can be applied to a wide class of continuous-time systems, and it does not require computation of Lyapunov exponents in order to check synchronization. In addition, note that the authors have successfully implemented in hardware a similar continuous-time observer (though without a scaling ability) [11], indicating that the observer-based method is able to be electronically realized.
3. Examples 3.1. Rössler’s system The dynamics of Rössler’s system [26] can be written in the form (3) as:
32 3 2 3 2 3 2 3 x1 ðtÞ 0 1 1 0 0 0 x_ 1 ðtÞ 7 7 6 7 6 x_ ðtÞ 7 6 1 0:25 607 6 0 1 0 x ðtÞ 2 76 7 6 7 6 2 7 6 6 7 76 7¼6 7 þ 6 7x1 ðtÞx3 ðtÞ þ 6 7: 6 4 x_ 3 ðtÞ 5 4 0 435 0 0 0 54 x3 ðtÞ 5 4 1 5 _x4 ðtÞ 0 0 0:5 0:05 0 x4 ðtÞ 0 2
ð11Þ
System (11) exhibits hyperchaotic behavior when the initial conditions are given by x1(0) = 20, x2(0) = x3(0) = 0 and x4(0) = 15 [26]. Since matrix (5) is full rank, the error system eigenvalues can be moved anywhere by Theorem 1. By placing them at 1, the output of (11) becomes the scalar signal
yðtÞ ¼ x1 ðtÞx3 ðtÞ þ ½ 3:3712 0:9561 4:3 5:8126 ½ x1 ðtÞ x2 ðtÞ x3 ðtÞ x4 ðtÞ T :
ð12Þ
According to (6) and (7), the dynamics of the nonlinear observer are described by
3 2 ^x_ 1 ðtÞ a11 7 6 ^_ a21 6 x2 ðtÞ 7 6 6 ¼ 7 6_ 4 ^x3 ðtÞ 5 4 a31 a41 ^x_ 4 ðtÞ 2 2
a13 a23 a33 a43
3 2
32
a14 1 0 1 1 0 a11 6 6 a24 7 0 1 7 7 6 1 0:25 76 a21 0 0 54 a31 a34 5 4 0 0 a44 a41 0 0 0:5 0:05 3 2 3
2
a12 a22 a32 a42
a13 a23 a33 a43
a14 1 0 a11 6 6 7 a24 7 7 6 0 7x1 ðtÞx3 ðtÞ þ 6 a21 4 a31 a34 5 4 1 5 a44 a41 0
a11 a12 a22 a32 a41 a42
a13 a23 a33 a43
a14 1 0 6 7 a24 7 7 607 ^ a34 5 4 1 5ðyðtÞ yðtÞÞ; a44 0
a11 6 a21 6 þ4 a31 a41 2
a12 a22 a32 a42
6 a21 þ6 4 a31
a12 a22 a32 a42
a12 a22 a32 a42 a13 a23 a33 a43
a13 a23 a33 a43
32
3
a14 ^x1 ðtÞ 6 7 a24 7 76 ^x2 ðtÞ 7 a34 54 ^x3 ðtÞ 5 a44 ^x4 ðtÞ
3 2 3
a14 1 0 6 7 a24 7 7 607 a34 5 4 3 5 a44 0
3 2 3
ð13Þ
2
a11 6a 21 ^ðtÞ ¼ ^x1 ðtÞ^x3 ðtÞ þ ½ 3:3712 0:9561 4:3 5:8126 6 y 6 4 a31 a41
a12 a22 a32 a42
a13 a23 a33 a43
32
3
a14 ^x1 ðtÞ 7 6 a24 7 76 ^x2 ðtÞ 7 7: 76 5 4 ^ a34 x3 ðtÞ 5 a44 ^x4 ðtÞ
ð14Þ
G. Grassi, D.A. Miller / Applied Mathematics and Computation 218 (2012) 6118–6124
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According to Theorem 1, the response system (13) and the drive system (11) are synchronized for any value of the scaling factors aij. In order to show the effectiveness of the approach, some simulations have been carried using the following scaling matrix:
2
1 3 0 4
3
60 2 0 07 7 7: 40 5 1 35
a¼6 6
0
0
ð15Þ
0 2
By taking into account the error (2) and the matrix (15), it follows that kðx1 ðtÞ ^ x1 ðtÞ 3^ x2 ðtÞ 4^ x4 ðtÞÞk ! 0, i.e., the drive system state x1(t) synchronizes with the linear combination of response system states given by ð^ x1 ðtÞ þ 3^ x2 ðtÞ þ 4^ x4 ðtÞÞ. Moreover, kðx2 ðtÞ 2^ x2 ðtÞÞk ! 0, i.e., the response system state ^ x2 ðtÞ is scaled by half with respect to the drive system state x2(t). The dynamic behavior of the drive-response system pair can be observed in Fig. 1. In particular, Fig. 1 shows on the left the attractor of the drive system in the (x1,x2)-plane. This plot clearly highlights the well-known shape of the Rössler’s attractor. On other hand, Fig. 1 shows on the right the attractor of the response system in the ð^ x1 ; ^ x2 Þ-plane. The shape of this attractor is completely different from that of the drive system attractor, by virtue of the fact that each drive system state synchronizes with a linear combination of the response system states. Similarly, by taking into account (2) and (15), it follows that kðx3 ðtÞ 5^ x2 ðtÞ ^ x3 ðtÞ 3^ x4 ðtÞÞk ! 0, i.e., the drive system state x3(t) synchronizes with the linear combination of response system states given by ð5^ x2 ðtÞ þ ^ x3 ðtÞ þ 3^ x4 ðtÞÞ. Fig. 2 depicts the drive system attractor in the (x2, x3)-plane (left) and the response system attractor in the ð^ x2 ; ^ x3 Þ-plane (right). Note that the shape of response system attractor is completely different from that of the drive system attractor. In particular, the dynamic range of the state x3(t) ranges from 0 to 250, whereas the dynamic range of ^ x3 ðtÞ ranges from 200 to 250. In spite of the different shapes of the phase plots, the drive and response systems are synchronized by virtue of Theorem 1.
Fig. 1. Attractor of the drive system (11) in the (x1, x2)-plane (left) and attractor of the response system (13) in the ð^ x1 ; ^ x2 Þ-plane (right). In order to compare the sizes and the shapes of the attractors, note that the two plots have the same scale. The scaling matrix is given in (15).
Fig. 2. Attractor of the drive system (11) in the (x2, x3)-plane (left) and attractor of the response system (13) in the ð^ x2 ; ^ x3 Þ-plane (right). In order to compare the sizes and the shapes of the attractors, note that the two plots have the same scale. The scaling matrix is given in (15).
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3.2. Hyperchaotic oscillator The drive system considered here is the hyperchaotic oscillator implemented in [11]. The circuit dynamics can be written in dimensionless form (3) as in [8] for the case of ideal (zero resistance) inductors:
3 32 3 2 3 2 x1 ðtÞ 0 0:7 1 1 0 x_ 1 ðtÞ 7 6 7 6 6 x_ ðtÞ 7 6 1 0 0 0 7 76 x2 ðtÞ 7 6 0 7 6 2 7 6 7ðx4 ðtÞ 1ÞHðx4 ðtÞ 1Þ: 76 7þ6 6 7¼6 4 x_ 3 ðtÞ 5 4 3 0 0 3 54 x3 ðtÞ 5 4 0 5 30 0 0 3 0 x4 ðtÞ x_ 4 ðtÞ 2
ð16Þ
where H(z) is the Heaviside function, i.e., H(z < 0) = 0 and H(z P 0) = 1. Since matrix (5) is full rank, the error system eigenvalues can be moved anywhere by Theorem 1. By placing them at 1, the output of (16) becomes the scalar signal
yðtÞ ¼ ðx4 ðtÞ 1ÞHðx4 ðtÞ 1Þ þ ½ 47:49 104:17 6:83 1:35 ½ x1 ðtÞ x2 ðtÞ x3 ðtÞ x4 ðtÞ T
ð17Þ
whereas the dynamics of the nonlinear observer are described by
3 2 ^x_ 1 ðtÞ a11 7 6 6 ^x_ ðtÞ 7 6 a 6 2 7 6 21 7¼6 6 6 ^x_ 3 ðtÞ 7 4 a31 5 4 a41 ^x_ 4 ðtÞ 2 2
a11 6a 6 21 þ6 4 a31 a41
a12 a22 a32 a42
a12 a22 a32 a42 a13 a23 a33 a43
a13 a23 a33 a43
32
3 2
a14 1 0:7 1 1 0 a11 7 7 6 6 a24 7 6 1 0 0 0 76 a21 76 7 6 a34 5 4 3 0 0 3 54 a31 0 0 3 0 a44 a41
3 2
3
a12 a22 a32 a42
2
a14 1 0 a11 6 0 7 6a a24 7 7 7 6 6 21 7ðx ðtÞ 1ÞHðx4 ðtÞ 1Þ þ 6 7 6 4 a31 a34 5 4 0 5 4 30 a44 a41 2
a11 6a 21 ^ðtÞ ¼ ½ 47:49 104:17 6:83 1:35 6 y 6 4 a31 a41
a12 a22 a32 a42
a13 a23 a33 a43
32
a13 a23 a33 a43 a12 a22 a32 a42
32
3
a14 ^x1 ðtÞ 7 6 a24 7 76 ^x2 ðtÞ 7 7 76 5 4 ^ a34 x3 ðtÞ 5 a44 ^x4 ðtÞ a13 a23 a33 a43
3 2
3
a14 1 0 7 6 a24 7 7 6 0 7 ^ðtÞÞ; 7ðyðtÞ y 7 6 5 4 0 5 a34 30 a44
ð18Þ
3
a14 ^x1 ðtÞ 7 6 a24 7 76 ^x2 ðtÞ 7 7 þ ð^x4 ðtÞ 1ÞHð^x4 ðtÞ 1Þ: 76 5 4 ^ a34 x3 ðtÞ 5 a44 ^x4 ðtÞ
ð19Þ
The response system (18) and the drive system (16) are synchronized for any scaling matrix (1). In order to show the effectiveness of the approach, some simulations have been carried out by choosing the following scaling matrix:
0
0
3
6 0 1 0 40 0 2
0 0
7 7 7: 5
2
2
0
a¼6 6
0
0
ð20Þ
0 1=2
Since in this case a is a diagonal matrix with different entries, full state hybrid projective synchronization will be achieved. In particular, Fig. 3 shows the attractor of the drive system (16) in the (x1, x2)-plane (left) and the attractor of the response sys-
Fig. 3. Attractor of the drive system (16) in the (x1, x2)-plane (left) and attractor of the response system (18) in the ð^ x1 ; ^ x2 Þ-plane (right). The scaling matrix is given in (20). Compare the attractors by noting that the two plots have the same scale: ^ x1 ðtÞ has been scaled by half, whereas ^ x2 ðtÞ has been reflected.
G. Grassi, D.A. Miller / Applied Mathematics and Computation 218 (2012) 6118–6124
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Fig. 4. Attractor of the drive system (16) in the (x3, x4)-plane (left) and attractor of the observer (18) in the ð^ x3 ; ^ x4 Þ-plane (right). The scaling matrix is given in (20). Compare the attractors by noting that the plots have the same scale: ^ x3 ðtÞ has been scaled by half, whereas ^ x4 ðtÞ has been scaled by two.
tem (18) in the ð^ x1 ; ^ x2 Þ-plane (right). These two attractors, which are synchronized by virtue of Theorem 1, have different sizes, since the variable ^ x1 ðtÞ has been scaled by half (a11 = 2) whereas the variable ^ x2 ðtÞ has been reflected (a22 = 1). Moreover, Fig. 4 shows the attractor of the drive system (16) in the (x3, x4)-plane (left) and the attractor of the response system (18) in the ð^ x3 ; ^ x4 Þ-plane (right). These synchronized attractors have different shapes, since the variable ^ x3 ðtÞ has been scaled by half (a33 = 2) whereas the variable ^ x4 ðtÞ has been scaled by two (a44 = 1/2). Note that as a subsystem the variables ^ x1 ðtÞ; ^ x3 ðtÞ and ^ x4 ðtÞ achieve projective synchronization with the corresponding drive system variables x1(t), x3(t) and x4(t), respectively, whereas the variable ^ x2 ðtÞ is in anti-phase synchronization with the variable x2(t). 4. Conclusion This paper has introduced a novel type of synchronization, where each drive system state synchronizes with a linear combination of response system states. A remarkable feature of the proposed observer-based approach is that synchronization is achievable via a scalar signal. The method, which has proved to be rigorous and systematic, is based on a theorem that assure asymptotic synchronization for a wide class of chaotic (hyperchaotic) systems. Two examples, involving Rössler’s system and a hyperchaotic oscillator, have clearly shown that the proposed technique is a general framework to achieve any type of synchronization defined to date. In conclusion, as far as the authors are aware, the method developed herein is the only one to simultaneously include the following features: (i) synchronization is achieved up to an arbitrary scaling matrix; (ii) synchronization is achieved using a scalar signal; (iii) it is rigorous (being based on a theorem); and (iv) it can be applied to a wide class of chaotic (hyperchaotic) systems. Acknowledgments The co-author Giuseppe Grassi would like to thank the Western Michigan University Department of Electrical and Computer Engineering for the hospitality during summer 2011. References [1] T.L. Carroll, L.M. Pecora, Synchronizing chaotic circuits, IEEE Transactions on Circuits and Systems 38 (4) (1991) 453–456. [2] M. Ogorzalek, Taming chaos – Part I: Synchronization, IEEE Transactions on Circuits and Systems I 40 (1993) 693–699. [3] M. Brucoli, L. Carnimeo, G. Grassi, A method for the synchronization of hyperchaotic circuits, International Journal of Bifurcation Chaos 6 (9) (1996) 1673–1681. [4] Congxu Zhu, Control and synchronize a novel hyperchaotic system, Applied Mathematics and Computation 216 (1) (2010) 276–284. [5] G. Grassi, S. Mascolo, A system theory approach for designing cryptosystems based on hyperchaos, IEEE Transactions on Circuits and Systems – I: Fundamental Theory and Applications 46 (9) (1999) 1135–1138. [6] G. Grassi, S. Mascolo, Synchronization of high-order oscillators by observer design with application to hyperchaos-based cryptography, International Journal of Circuit Theory and Applications 27 (6) (1999) 543–553. [7] G. Grassi, S. Mascolo, Nonlinear observer design to synchronize hyperchaotic systems via a scalar signal, IEEE Transactions on Circuits and Systems – I: Fundamental Theory and Applications 44 (10) (1997) 1011–1014. [8] G. Grassi, S. Mascolo, Synchronizing hyperchaotic systems by observer design, IEEE Transactions on Circuits and Systems – II: Analog and Digital Signal Processing 46 (4) (1999) 478–483. [9] X.F. Wang, Z.Q. Wang, Synchronizing chaos and hyperchaos with any scalar transmitted signal, IEEE Transactions on Circuits and Systems – I: Fundamental Theory and Applications 45 (10) (1998) 1101–1103. [10] G. Grassi, S. Mascolo, Synchronisation of hyperchaotic oscillators using a scalar signal, IEE Electronics Letters 34 (5) (1998) 424–425. [11] D.A. Miller, G. Grassi, Experimental realization of observer-based hyperchaos synchronization, IEEE Transactions on Circuits and Systems – I: Fundamental Theory and Applications 48 (3) (2001) 366–374. [12] G. Grassi, D.A. Miller, Theory and experimental realization of observer-based discrete-time hyperchaos synchronization, IEEE Transactions on Circuits and Systems – I: Fundamental Theory and Applications 49 (3) (2002) 373–378.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Synchronizing chaotic systems up to an arbitrary scaling matrix via a single signal Giuseppe Grassi a,⇑, Damon A. Miller b a b
Dipartimento Ingegneria Innovazione, Università del Salento, 73100 Lecce, Italy Department of Electrical and Computer Engineering, Western Michigan University, Kalamazoo, MI 49008, USA
a r t i c l e
i n f o
Keywords: Chaos synchronization Observer-based synchronization Chaotic systems with attractor scaling Projective synchronization Full state hybrid projective synchronization
a b s t r a c t This paper introduces a novel type of synchronization, where two chaotic systems synchronize up to an arbitrary scaling matrix. In particular, each drive system state synchronizes with a linear combination of response system states by using a single synchronizing signal. The proposed observer-based method exploits a theorem that assures asymptotic synchronization for a wide class of continuous-time chaotic (hyperchaotic) systems. Two examples, involving Rössler’s system and a hyperchaotic oscillator, show that the proposed technique is a general framework to achieve any type of synchronization defined to date. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The concept of chaos synchronization has attracted considerable attention in recent years. In 1991 Carroll and Pecora [1] showed that the dynamics of a drive system and of a driven subsystem (response system) become synchronized if the Lyapunov exponents of the response system are less then zero. Since their work, several methods for obtaining synchronized systems (including neural networks and complex systems) have been developed [2–14]. Among these, some researchers have focused their attention on the synchronization of hyperchaotic systems using a scalar synchronizing signal [4–10]. A particular type of chaos synchronization was introduced in 1999 [15]. In this type of synchronization (called projective synchronization) the response system states are scaled replicas of the drive system states [16]. Since the scaling factor was not predictable in [15], some control methods have been successively proposed, with the aim to choose any desired scaling factor [16–20]. A variation of project synchronization is the idea of full state hybrid projective synchronization, which has been recently developed in [21–24]. In this type of synchronization the scaling factor can be different for each state variable. This enables different kinds of synchronization to be achieved. For example, in Ref. [21] anti-synchronization and complete synchronization are simultaneously achieved in a symmetrical coordinate subspace and a normal coordinate subspace, respectively. In [22] adaptive control is exploited to achieve the full state hybrid projective synchronization of a new 4D hyperchaotic system with unknown parameters. Additionally, in [23,24] an active control scheme is utilized to achieve full state hybrid projective synchronization in some hyperchaotic continuous-time systems. The approaches in [21–24] share the drawback that synchronization is achieved using several control signals, while communications based on projective synchronization require a scalar synchronizing signal [25]. This paper presents a further contribution to the topic of synchronization by introducing a novel scheme, where two chaotic systems synchronize up to an arbitrary scaling matrix. Specifically, each drive system state synchronizes with a linear
⇑ Corresponding author. E-mail address: [email protected] (G. Grassi). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.097
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combination of response system states by using a single synchronizing signal. The proposed approach proves to be a general framework, which includes (as a particular case) any type of synchronization defined to date. The paper is organized as follows. Section 2 introduces the proposed observer-based approach, along with a theorem that guarantees synchronization can be achieved for any arbitrary scaling matrix under certain broad conditions. A remarkable feature is that the proposed type of synchronization is achievable via a single signal for a wide class of continuous-time chaotic systems. Finally, Section 3 illustrates two examples, involving Rössler’s system [26] and the hyperchaotic oscillator in [11], showing that identical, projective, anti-phase and full state hybrid projective synchronization are all achievable using the conceived synchronization scheme. 2. Synchronizing chaotic systems up to an arbitrary scaling matrix Definition 1. Given the continuous-time drive and response systems described by the n-dimensional state vectors x(t) and ^ðtÞ, respectively, these two systems are synchronized up to an arbitrary scaling matrix when, for an initial condition x ^ð0Þ, there x is a matrix
2
3
a11 a12 a1n a22 a2n 7 7
6a 21 a¼6 6 4
72R 5
nn
ð1Þ
an1 an2 ann such that:
! n X kei ðtÞk ¼ xi ðtÞ aij ^xj ðtÞ ! 0 as t ! 1; j¼1
i ¼ 1; 2; . . . n:
ð2Þ
The condition (2) means that each drive system variable xi(t) synchronizes with a linear combination of response system variables ðai1 ^ x1 ðtÞ þ ai2 ^ x2 ðtÞ þ ain ^ xn ðtÞÞ; i ¼ 1; 2; . . . n. Referring to the systems to be synchronized, this paper focuses on the class of dynamical systems defined by the following state and output equations, respectively:
_ xðtÞ ¼ AxðtÞ þ bf ðxðtÞÞ þ c;
ð3Þ
yðtÞ ¼ f ðxðtÞÞ þ kxðtÞ;
ð4Þ
n1
nn
n1
n1
n
where xðtÞ 2 R is the state vector, A 2 R ; b 2 R and c 2 R are constant matrices, f : R ! R is a scalar nonlinear function, y(t) is a scalar synchronizing output, and k 2 R1n is a gain vector to be determined. As pointed out in [7], the class described by (3) includes several well-known chaotic and hyperchaotic systems, such as Chua’s circuit [2], Rössler’s system [26], the Matsumoto–Chua–Kobayashi circuit [27], the hyperchaotic oscillator in [11], the higher dimensional Chua’s circuit in [28] and the circuit for generating multi-scroll attractors in [29]. Given the drive system (3), the observed-based approach enables the state x(t) to be reconstructed by a response system using a scalar function of the states (4). By utilizing the fact that the response system need not be a copy of the drive system in order to obtain synchronized dynamics (see [8,11]), a theorem is now proposed, where synchronization up to an arbitrary scaling matrix is achievable via a single signal. Theorem 1. If the matrix
b Ab A2 b An1 b
ð5Þ
is full rank, then the nonlinear dynamical system
^ðtÞÞ þ a1 c þ a1 bðyðtÞ y ^_ ðtÞ ¼ a1 Aax ^ðtÞ þ a1 bf ðx ^ðtÞÞ; x ^ ^ ^ yðtÞ ¼ f ðxðtÞÞ þ kaxðtÞ
ð6Þ ð7Þ
and the drive system (3) synchronize for any arbitrary scaling matrix a, provided that the eigenvalues of the matrix
½A bk
ð8Þ
are placed in the left-half plane by suitable k. Proof. First of all, it is worth noting that the error (2) can be written in matrix form as
^ðtÞÞk ! 0 keðtÞk ¼ kðxðtÞ ax By taking into account the error defined in (2), it follows that:
ð9Þ
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_ _ ax ^ðtÞ bf ðx ^ðtÞÞ c bðyðtÞ y ^_ ðtÞÞ ¼ AxðtÞ þ bf ðxðtÞÞ þ c Aax ^ðtÞÞ eðtÞ ¼ ðxðtÞ ^ðtÞÞ þ bf ðxðtÞÞ bf ðx ^ðtÞÞ bðf ðxðtÞÞ þ kxðtÞ f ðx ^ðtÞÞ kax ^ðtÞÞ ¼ AðxðtÞ ax ^ðtÞÞ bkðxðtÞ ax ^ðtÞÞ ¼ Ae bke ¼ ðA bkÞeðtÞ ¼ AðxðtÞ ax
ð10Þ
_ If matrix (5) is full rank, the error system eðtÞ ¼ ðA bkÞeðtÞ can be globally asymptotically stabilized at the origin by suit^ð0Þ and for any arbitrary scaling matrix a. Thus, according to (2) able k [7,8]. This means that ke(t)k ? 0 for t ? 1, for any x each drive system state synchronizes with a linear combination of response system states via the scalar synchronizing signal (4). Remark 1. The proposed scheme represents a general framework to achieve any type of synchronization defined to date. Namely, when a11 = a22 = = ann = 1 and aij = 0 i – j, identical (complete) synchronization is achieved, whereas for a11 = a22 = = ann = 1 and aij = 0 i – j, anti-phase synchronization is obtained. Moreover, when a11 = a22 = = ann = a and aij = 0 i – j, the projective synchronization as originally proposed in [15] is obtained. On the other hand, when a11 – a22 – – ann and aij = 0 i – j, the scaling matrix a is diagonal with different entries, and therefore full state hybrid projective synchronization as defined in [24] is achieved. Remark 2. The proposed observer-based method shares the same advantages illustrated in [8], i.e., it is rigorous and systematic, it can be applied to a wide class of continuous-time systems, and it does not require computation of Lyapunov exponents in order to check synchronization. In addition, note that the authors have successfully implemented in hardware a similar continuous-time observer (though without a scaling ability) [11], indicating that the observer-based method is able to be electronically realized.
3. Examples 3.1. Rössler’s system The dynamics of Rössler’s system [26] can be written in the form (3) as:
32 3 2 3 2 3 2 3 x1 ðtÞ 0 1 1 0 0 0 x_ 1 ðtÞ 7 7 6 7 6 x_ ðtÞ 7 6 1 0:25 607 6 0 1 0 x ðtÞ 2 76 7 6 7 6 2 7 6 6 7 76 7¼6 7 þ 6 7x1 ðtÞx3 ðtÞ þ 6 7: 6 4 x_ 3 ðtÞ 5 4 0 435 0 0 0 54 x3 ðtÞ 5 4 1 5 _x4 ðtÞ 0 0 0:5 0:05 0 x4 ðtÞ 0 2
ð11Þ
System (11) exhibits hyperchaotic behavior when the initial conditions are given by x1(0) = 20, x2(0) = x3(0) = 0 and x4(0) = 15 [26]. Since matrix (5) is full rank, the error system eigenvalues can be moved anywhere by Theorem 1. By placing them at 1, the output of (11) becomes the scalar signal
yðtÞ ¼ x1 ðtÞx3 ðtÞ þ ½ 3:3712 0:9561 4:3 5:8126 ½ x1 ðtÞ x2 ðtÞ x3 ðtÞ x4 ðtÞ T :
ð12Þ
According to (6) and (7), the dynamics of the nonlinear observer are described by
3 2 ^x_ 1 ðtÞ a11 7 6 ^_ a21 6 x2 ðtÞ 7 6 6 ¼ 7 6_ 4 ^x3 ðtÞ 5 4 a31 a41 ^x_ 4 ðtÞ 2 2
a13 a23 a33 a43
3 2
32
a14 1 0 1 1 0 a11 6 6 a24 7 0 1 7 7 6 1 0:25 76 a21 0 0 54 a31 a34 5 4 0 0 a44 a41 0 0 0:5 0:05 3 2 3
2
a12 a22 a32 a42
a13 a23 a33 a43
a14 1 0 a11 6 6 7 a24 7 7 6 0 7x1 ðtÞx3 ðtÞ þ 6 a21 4 a31 a34 5 4 1 5 a44 a41 0
a11 a12 a22 a32 a41 a42
a13 a23 a33 a43
a14 1 0 6 7 a24 7 7 607 ^ a34 5 4 1 5ðyðtÞ yðtÞÞ; a44 0
a11 6 a21 6 þ4 a31 a41 2
a12 a22 a32 a42
6 a21 þ6 4 a31
a12 a22 a32 a42
a12 a22 a32 a42 a13 a23 a33 a43
a13 a23 a33 a43
32
3
a14 ^x1 ðtÞ 6 7 a24 7 76 ^x2 ðtÞ 7 a34 54 ^x3 ðtÞ 5 a44 ^x4 ðtÞ
3 2 3
a14 1 0 6 7 a24 7 7 607 a34 5 4 3 5 a44 0
3 2 3
ð13Þ
2
a11 6a 21 ^ðtÞ ¼ ^x1 ðtÞ^x3 ðtÞ þ ½ 3:3712 0:9561 4:3 5:8126 6 y 6 4 a31 a41
a12 a22 a32 a42
a13 a23 a33 a43
32
3
a14 ^x1 ðtÞ 7 6 a24 7 76 ^x2 ðtÞ 7 7: 76 5 4 ^ a34 x3 ðtÞ 5 a44 ^x4 ðtÞ
ð14Þ
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According to Theorem 1, the response system (13) and the drive system (11) are synchronized for any value of the scaling factors aij. In order to show the effectiveness of the approach, some simulations have been carried using the following scaling matrix:
2
1 3 0 4
3
60 2 0 07 7 7: 40 5 1 35
a¼6 6
0
0
ð15Þ
0 2
By taking into account the error (2) and the matrix (15), it follows that kðx1 ðtÞ ^ x1 ðtÞ 3^ x2 ðtÞ 4^ x4 ðtÞÞk ! 0, i.e., the drive system state x1(t) synchronizes with the linear combination of response system states given by ð^ x1 ðtÞ þ 3^ x2 ðtÞ þ 4^ x4 ðtÞÞ. Moreover, kðx2 ðtÞ 2^ x2 ðtÞÞk ! 0, i.e., the response system state ^ x2 ðtÞ is scaled by half with respect to the drive system state x2(t). The dynamic behavior of the drive-response system pair can be observed in Fig. 1. In particular, Fig. 1 shows on the left the attractor of the drive system in the (x1,x2)-plane. This plot clearly highlights the well-known shape of the Rössler’s attractor. On other hand, Fig. 1 shows on the right the attractor of the response system in the ð^ x1 ; ^ x2 Þ-plane. The shape of this attractor is completely different from that of the drive system attractor, by virtue of the fact that each drive system state synchronizes with a linear combination of the response system states. Similarly, by taking into account (2) and (15), it follows that kðx3 ðtÞ 5^ x2 ðtÞ ^ x3 ðtÞ 3^ x4 ðtÞÞk ! 0, i.e., the drive system state x3(t) synchronizes with the linear combination of response system states given by ð5^ x2 ðtÞ þ ^ x3 ðtÞ þ 3^ x4 ðtÞÞ. Fig. 2 depicts the drive system attractor in the (x2, x3)-plane (left) and the response system attractor in the ð^ x2 ; ^ x3 Þ-plane (right). Note that the shape of response system attractor is completely different from that of the drive system attractor. In particular, the dynamic range of the state x3(t) ranges from 0 to 250, whereas the dynamic range of ^ x3 ðtÞ ranges from 200 to 250. In spite of the different shapes of the phase plots, the drive and response systems are synchronized by virtue of Theorem 1.
Fig. 1. Attractor of the drive system (11) in the (x1, x2)-plane (left) and attractor of the response system (13) in the ð^ x1 ; ^ x2 Þ-plane (right). In order to compare the sizes and the shapes of the attractors, note that the two plots have the same scale. The scaling matrix is given in (15).
Fig. 2. Attractor of the drive system (11) in the (x2, x3)-plane (left) and attractor of the response system (13) in the ð^ x2 ; ^ x3 Þ-plane (right). In order to compare the sizes and the shapes of the attractors, note that the two plots have the same scale. The scaling matrix is given in (15).
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3.2. Hyperchaotic oscillator The drive system considered here is the hyperchaotic oscillator implemented in [11]. The circuit dynamics can be written in dimensionless form (3) as in [8] for the case of ideal (zero resistance) inductors:
3 32 3 2 3 2 x1 ðtÞ 0 0:7 1 1 0 x_ 1 ðtÞ 7 6 7 6 6 x_ ðtÞ 7 6 1 0 0 0 7 76 x2 ðtÞ 7 6 0 7 6 2 7 6 7ðx4 ðtÞ 1ÞHðx4 ðtÞ 1Þ: 76 7þ6 6 7¼6 4 x_ 3 ðtÞ 5 4 3 0 0 3 54 x3 ðtÞ 5 4 0 5 30 0 0 3 0 x4 ðtÞ x_ 4 ðtÞ 2
ð16Þ
where H(z) is the Heaviside function, i.e., H(z < 0) = 0 and H(z P 0) = 1. Since matrix (5) is full rank, the error system eigenvalues can be moved anywhere by Theorem 1. By placing them at 1, the output of (16) becomes the scalar signal
yðtÞ ¼ ðx4 ðtÞ 1ÞHðx4 ðtÞ 1Þ þ ½ 47:49 104:17 6:83 1:35 ½ x1 ðtÞ x2 ðtÞ x3 ðtÞ x4 ðtÞ T
ð17Þ
whereas the dynamics of the nonlinear observer are described by
3 2 ^x_ 1 ðtÞ a11 7 6 6 ^x_ ðtÞ 7 6 a 6 2 7 6 21 7¼6 6 6 ^x_ 3 ðtÞ 7 4 a31 5 4 a41 ^x_ 4 ðtÞ 2 2
a11 6a 6 21 þ6 4 a31 a41
a12 a22 a32 a42
a12 a22 a32 a42 a13 a23 a33 a43
a13 a23 a33 a43
32
3 2
a14 1 0:7 1 1 0 a11 7 7 6 6 a24 7 6 1 0 0 0 76 a21 76 7 6 a34 5 4 3 0 0 3 54 a31 0 0 3 0 a44 a41
3 2
3
a12 a22 a32 a42
2
a14 1 0 a11 6 0 7 6a a24 7 7 7 6 6 21 7ðx ðtÞ 1ÞHðx4 ðtÞ 1Þ þ 6 7 6 4 a31 a34 5 4 0 5 4 30 a44 a41 2
a11 6a 21 ^ðtÞ ¼ ½ 47:49 104:17 6:83 1:35 6 y 6 4 a31 a41
a12 a22 a32 a42
a13 a23 a33 a43
32
a13 a23 a33 a43 a12 a22 a32 a42
32
3
a14 ^x1 ðtÞ 7 6 a24 7 76 ^x2 ðtÞ 7 7 76 5 4 ^ a34 x3 ðtÞ 5 a44 ^x4 ðtÞ a13 a23 a33 a43
3 2
3
a14 1 0 7 6 a24 7 7 6 0 7 ^ðtÞÞ; 7ðyðtÞ y 7 6 5 4 0 5 a34 30 a44
ð18Þ
3
a14 ^x1 ðtÞ 7 6 a24 7 76 ^x2 ðtÞ 7 7 þ ð^x4 ðtÞ 1ÞHð^x4 ðtÞ 1Þ: 76 5 4 ^ a34 x3 ðtÞ 5 a44 ^x4 ðtÞ
ð19Þ
The response system (18) and the drive system (16) are synchronized for any scaling matrix (1). In order to show the effectiveness of the approach, some simulations have been carried out by choosing the following scaling matrix:
0
0
3
6 0 1 0 40 0 2
0 0
7 7 7: 5
2
2
0
a¼6 6
0
0
ð20Þ
0 1=2
Since in this case a is a diagonal matrix with different entries, full state hybrid projective synchronization will be achieved. In particular, Fig. 3 shows the attractor of the drive system (16) in the (x1, x2)-plane (left) and the attractor of the response sys-
Fig. 3. Attractor of the drive system (16) in the (x1, x2)-plane (left) and attractor of the response system (18) in the ð^ x1 ; ^ x2 Þ-plane (right). The scaling matrix is given in (20). Compare the attractors by noting that the two plots have the same scale: ^ x1 ðtÞ has been scaled by half, whereas ^ x2 ðtÞ has been reflected.
G. Grassi, D.A. Miller / Applied Mathematics and Computation 218 (2012) 6118–6124
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Fig. 4. Attractor of the drive system (16) in the (x3, x4)-plane (left) and attractor of the observer (18) in the ð^ x3 ; ^ x4 Þ-plane (right). The scaling matrix is given in (20). Compare the attractors by noting that the plots have the same scale: ^ x3 ðtÞ has been scaled by half, whereas ^ x4 ðtÞ has been scaled by two.
tem (18) in the ð^ x1 ; ^ x2 Þ-plane (right). These two attractors, which are synchronized by virtue of Theorem 1, have different sizes, since the variable ^ x1 ðtÞ has been scaled by half (a11 = 2) whereas the variable ^ x2 ðtÞ has been reflected (a22 = 1). Moreover, Fig. 4 shows the attractor of the drive system (16) in the (x3, x4)-plane (left) and the attractor of the response system (18) in the ð^ x3 ; ^ x4 Þ-plane (right). These synchronized attractors have different shapes, since the variable ^ x3 ðtÞ has been scaled by half (a33 = 2) whereas the variable ^ x4 ðtÞ has been scaled by two (a44 = 1/2). Note that as a subsystem the variables ^ x1 ðtÞ; ^ x3 ðtÞ and ^ x4 ðtÞ achieve projective synchronization with the corresponding drive system variables x1(t), x3(t) and x4(t), respectively, whereas the variable ^ x2 ðtÞ is in anti-phase synchronization with the variable x2(t). 4. Conclusion This paper has introduced a novel type of synchronization, where each drive system state synchronizes with a linear combination of response system states. A remarkable feature of the proposed observer-based approach is that synchronization is achievable via a scalar signal. The method, which has proved to be rigorous and systematic, is based on a theorem that assure asymptotic synchronization for a wide class of chaotic (hyperchaotic) systems. Two examples, involving Rössler’s system and a hyperchaotic oscillator, have clearly shown that the proposed technique is a general framework to achieve any type of synchronization defined to date. In conclusion, as far as the authors are aware, the method developed herein is the only one to simultaneously include the following features: (i) synchronization is achieved up to an arbitrary scaling matrix; (ii) synchronization is achieved using a scalar signal; (iii) it is rigorous (being based on a theorem); and (iv) it can be applied to a wide class of chaotic (hyperchaotic) systems. Acknowledgments The co-author Giuseppe Grassi would like to thank the Western Michigan University Department of Electrical and Computer Engineering for the hospitality during summer 2011. References [1] T.L. Carroll, L.M. Pecora, Synchronizing chaotic circuits, IEEE Transactions on Circuits and Systems 38 (4) (1991) 453–456. [2] M. Ogorzalek, Taming chaos – Part I: Synchronization, IEEE Transactions on Circuits and Systems I 40 (1993) 693–699. [3] M. Brucoli, L. Carnimeo, G. Grassi, A method for the synchronization of hyperchaotic circuits, International Journal of Bifurcation Chaos 6 (9) (1996) 1673–1681. [4] Congxu Zhu, Control and synchronize a novel hyperchaotic system, Applied Mathematics and Computation 216 (1) (2010) 276–284. [5] G. Grassi, S. Mascolo, A system theory approach for designing cryptosystems based on hyperchaos, IEEE Transactions on Circuits and Systems – I: Fundamental Theory and Applications 46 (9) (1999) 1135–1138. [6] G. Grassi, S. Mascolo, Synchronization of high-order oscillators by observer design with application to hyperchaos-based cryptography, International Journal of Circuit Theory and Applications 27 (6) (1999) 543–553. [7] G. Grassi, S. Mascolo, Nonlinear observer design to synchronize hyperchaotic systems via a scalar signal, IEEE Transactions on Circuits and Systems – I: Fundamental Theory and Applications 44 (10) (1997) 1011–1014. [8] G. Grassi, S. Mascolo, Synchronizing hyperchaotic systems by observer design, IEEE Transactions on Circuits and Systems – II: Analog and Digital Signal Processing 46 (4) (1999) 478–483. [9] X.F. Wang, Z.Q. Wang, Synchronizing chaos and hyperchaos with any scalar transmitted signal, IEEE Transactions on Circuits and Systems – I: Fundamental Theory and Applications 45 (10) (1998) 1101–1103. [10] G. Grassi, S. Mascolo, Synchronisation of hyperchaotic oscillators using a scalar signal, IEE Electronics Letters 34 (5) (1998) 424–425. [11] D.A. Miller, G. Grassi, Experimental realization of observer-based hyperchaos synchronization, IEEE Transactions on Circuits and Systems – I: Fundamental Theory and Applications 48 (3) (2001) 366–374. [12] G. Grassi, D.A. Miller, Theory and experimental realization of observer-based discrete-time hyperchaos synchronization, IEEE Transactions on Circuits and Systems – I: Fundamental Theory and Applications 49 (3) (2002) 373–378.
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